Correlation dimension and phase space contraction via extreme value theory

TL;DR

This paper introduces a novel method linking extreme value theory with dynamical systems to efficiently estimate key quantities like correlation dimension, entropy, and Lyapunov exponents through a new index called the Dynamical Extreme Index.

Contribution

It proposes a mathematically rigorous, parameter-free, and computationally simple approach to derive dynamical invariants from chaotic systems using extreme value laws.

Findings

The Dynamical Extreme Index accurately estimates correlation dimension, entropy, and Lyapunov exponents.

The method avoids the curse of dimensionality and does not require additional parameters.

A numerical code for the index's computation is provided.

Abstract

This study uses the link between extreme value laws and dynamical systems theory to show that important dynamical quantities as the correlation dimension, the entropy and the Lyapunov exponents can be obtained by fitting observables computed along a trajectory of a chaotic systems. All this information is contained in a newly defined Dynamical Extreme Index. Besides being mathematically well defined, it is almost numerically effortless to get as i) it does not require the specification of any additional parameter (e.g. embedding dimension, decorrelation time); ii) it does not suffer from the so-called curse of dimensionality. A numerical code for its computation is provided.